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Does anyone know a proof or reference to the following result?

Suppose that $A, B$ are both $m \times n$ real matrices. Then for all $1 \leq k \leq \min\{m, n\}$, $$|\sigma_k(A) - \sigma_k(B)| \leq \|A - B\|.$$

I think these are called the Weyl inequalities, and I remember learning a proof of this result using the minimax characterization of these singular values but I can't reconstruct the proof. Anyone know it or have a reference to it?

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  • $\begingroup$ @QiaochuYuan Beautiful set of notes, by the way. You should make a comment and I could select it as the selected answer? $\endgroup$
    – Drew Brady
    May 16, 2018 at 8:19

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You can find the proof in this blog post on SVD; I worked it out as an exercise. It's a corollary of the more general inequality

$$\sigma_{k+\ell+1}(A + B) \le \sigma_{k+1}(A) + \sigma_{\ell+1}(B)$$

which I think is what "the Weyl inequalities" refers to, and which can be proven using the minimax characterization of singular values.

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